• Suppose we are interested in understanding receptive field properties of a particular neuron.

• We know from past research that – in the absence of all external inputs – this neuron has a baseline firing rate of 10 spikes per second on average.

• However, the baseline firing rate is not constant, but fluctuates around this average value.

• Suppose we know the Expected Value of the baseline firing rate of this neuron is 10 spikes per second.

• Let \(X\) to be the random variable that generates the observed number of spikes per second during the stimulus.

• We want to know if the neuron is stimulus-driven.

• That is, we want to know if the Expected Value of \(X\) is greater than 10 spikes per second.

• \(\mathbb{E}(X) > 10\)???

• \(\mu_X > 10\)???

• Null Universe: The neuron is not stimulus-driven. The Expected Value of \(X\) is 10 spikes per second.

• Alternative Universe: The neuron is stimulus-driven.

• We want to know which universe we are in.

• We write this as:

\[ \begin{align*} H_0 &: \mu_X = 10 \\ H_1 &: \mu_X > 10 \end{align*} \]

• The Null (\(H_0\)) universe is the universe where the neuron is not stimulus-driven.

• The Expected Value of \(X_{H0}\) is 10 spikes per second.

• Assume that the distribution of \(X_{H0}\) is Normal with mean 10 and standard deviation \(\sigma_X = 1\).

• \(X \sim \mathscr{N}(\mu_X=10, \sigma_X=1)\)

• Firing rate - i.e., \(X_{H0}\) – is continuous and can take any value between 0 and \(\infty\).

• \(X \sim \mathscr{N}(\mu_X=10, \sigma_X=1)\) is continuous but is defined for \((-\infty, \infty)\).

• firing rate cannot be negative but our model can be negative. This is a weakness of the model.

• However, \(X \sim \mathscr{N}(\mu_X=10, \sigma_X=1)\) assigns very near zero probability to negative values, so this model may not be too bad afterall.

• Stimulate 25 times and observe the spikes per second in each case.

• Remember that our experiment generates the spike time raster plots from the beginning of this slide deck.

• We had to wrangle the data to get the number of spikes per second from those plots.

• This is where your data.table skills would come in handy.

• Recall that our question is expressed as follows:

\[ \begin{align*} H_0 &: \mu_X = 10 \\ H_1 &: \mu_X > 10 \end{align*} \]

• Our raw data doesn’t immediately and unambiguously tell us about \(\mu_X\).

• We can estimate \(\mu_X\) by taking the sample mean of the observed firing rates.

• We denote estimate of \(\mu_X\) as \(\widehat{\mu_X}\).

• \(\widehat{\mu_X} = \bar{x}\)

• \(\bar{x} = \frac{1}{n} \sum_{i=1}^{n} x_i\)

• This is the same as asking: \(P(\bar{X}_{H0} > \bar{x}_{obs})\)

• How can we compute this probability?

• We know that we need area under the curve of the probability distribution given a continuous distribution.

• This may give us the idea on the next slide…

• Can you see what’s wrong with this approach?

• Suspense…

• The random variable that generates the raw data (\(X\)) is different from the random variable that generates the sample mean (\(\bar{X}\)).

• Since we are asking about how likely an observed sample mean is under the Null, we should be using the distribution of the sample mean.

• The shaded area gives us \(P(\bar{X}_{H0} > \bar{x}_{obs})\)

• This is called the p-value.

• We are interested in the probability that the Expected Value – i.e., the population mean of the firing rate is greater than 10 spikes per second.

• The inequality is baked into the our original question.

• If the p-value is very small then the Null is very unlikely and we reject it.

• But how small is very small?

• This is choice we have to make and we will discuss it more when we discuss inferential errors and statistical power.

• For now, it’s enough to say that in pychology and neuroscience a p-value of 0.05 is often used.

• This number is called the Type I error rate and is denoted by the symbol \(\alpha\).

• For this example, the p-value turns out to be:

## [1] 0.08729804

1. Specify the null and alternative hypotheses (\(H_0\) and \(H_1\)) in terms of a population parameter \(\theta\).

2. Specify the type I error rate – denoted by the symbol \(\alpha\) – you are willing to tolerate.

3. Specify the sample statistic \(\widehat{\theta}\) that you will use to estimate the population parameter \(\theta\) in step 1 and state how it is distributed under the assumption that \(H_0\) is true.

4. Obtain a random sample and use it to compute the sample statistic from step 3. Call this value \(\widehat{\theta}_{\text{obs}}\).

5. If \(\widehat{\theta}_{\text{obs}}\) or a more extreme outcome is very unlikely to occur under the assumption that \(H_0\) is true, then reject \(H_0\). Otherwise, do not reject \(H_0\).

Consider the following figure. Can a Normal test be conducted on the basis of the information provided to you in this figure or is something essential missing?

Let \(\theta\) be a population parameter of interest and suppose that we know the following about it’s estimator \(\hat{\theta}\):

• What is the expected value of \(\hat{\theta}\)?

• What is the observed value of the test statistic?

• What is the \(p\)-value of this test?

• What is the critical value of this test (i.e., the value that ensures that type I error is less than or equal to \(0.05\))?